Simplify; express your answer in exponential form. Assume $y\neq 0, x\neq 0$. $\dfrac{{(y^{2}x^{-3})^{-3}}}{{(y^{5}x^{-5})^{-1}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{2}x^{-3})^{-3} = (y^{2})^{-3}(x^{-3})^{-3}}$ On the left, we have ${y^{2}}$ to the exponent ${-3}$ . Now ${2 \times -3 = -6}$ , so ${(y^{2})^{-3} = y^{-6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{2}x^{-3})^{-3}}}{{(y^{5}x^{-5})^{-1}}} = \dfrac{{y^{-6}x^{9}}}{{y^{-5}x^{5}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{-6}x^{9}}}{{y^{-5}x^{5}}} = \dfrac{{y^{-6}}}{{y^{-5}}} \cdot \dfrac{{x^{9}}}{{x^{5}}} = y^{{-6} - {(-5)}} \cdot x^{{9} - {5}} = y^{-1}x^{4}$